3.1 Predefined and Non-Variable Sources of Directivity
3.2.1 Particular Case of a Coaxial Multiway Acoustic Source
For many decades now acoustics specialists have paid special attention to making acoustic sets that are closest to the theoretical ideal essentially characterized by:                1. A faithful rendering of the entire audible frequency band ranging from 20 Hz to 20 kHz. This characteristic corresponds to a axial flat amplitude and linear phase response throughout the audible frequency band.        2. A low rate of non-linear distortion.        3. The absence of diffraction problems related to any kind of geometrical disposition of the speakers constituting the source. These effects can furthermore come from a discontinuity at the edges of the acoustic set.        4. A homogenous and well-controlled directivity especially at the level of the overlap frequency bands between the different speakers of the acoustic system.        
In practice, the verification of all the above criteria comes up against problems of very high technological complexity. Indeed, the first two requirements can be met by the association of several speakers dedicated to the rendering of complementary frequency sub-bands. However, the last two points, including the one related to directivity, are related to the geometry of the acoustic set or to the spatial disposition of the different speakers constituting it. Indeed, since there is no speaker capable of reproducing the totality of the audio signal (20 Hz to 20 kHz), a traditional acoustic set is generally constituted by several speakers, each dedicated to one frequency sub-band or channel. Each speaker is designed to present the best performance in terms of the frequency sub-band that it reproduces. The terminology often adopted to distinguish the different transducers is the following:                woofer: frequency band below the audio domain (f<200 Hz);        low-mid speaker: sub-band below the central frequency band of the audio domain (200 Hz<f<800 Hz);        high-mid speaker: sub-band above the central frequency band of the audio domain (800 Hz<f<4000 Hz);        tweeter: frequency band above the audio domain (f>4000 Hz).        
The frequencies f1=200 Hz, f2=800 Hz and f3=4000 Hz are called cross-frequencies. They demarcate the frequency bands generally allocated to each transducer.
In the classic configuration of a multiway acoustic source, the speakers are spread out on the facade of an acoustic set. This disposition of the transducers (membranes) has a drawback in terms of radiation at the level of the overlap frequency bands, especially in the context of the near field (i.e. in the vicinity of the acoustic set), given the delays between the sound waves emitted by the different speakers and perceived by the listener. In order to overcome these drawbacks, one innovative solution consists in reducing the inter-transducer spacing by adopting a coaxial configuration in which the different speakers are mounted on a same axis (the term “coaxial speaker with several membranes” is also used). This enables a more consistent rendering of the waves coming from the different transducers, even in very near fields.
When setting up a coaxial multiway acoustic source, great importance must be given to the characteristics of directivity of this source. Indeed, these characteristics significantly influence the subjective evaluation of the source once it is placed in a room. As a consequence, the sizes and profiles of the different membranes, as well as the geometry of the acoustic set, must be chosen with great care in order to attain a predefined target directivity since it is essentially these parameters that will condition the directivity of the acoustic set in the frequency band. Indeed, the speakers used are generally acoustically adapted to one another, i.e. they are designed to work on complementary frequency bands on which the response curves show the least possible unevenness so as to prevent excessively marked differences that would imbalance the sounds emitted. Thus, for a coaxial multiway source, it is generally desired to have an index of directivity of the source with an affine linear progression as a function of the frequency with a slope of the order of 5 dB/octave.
In the overlap bands, in order to ensure the continuity of the progress of the index of directivity, digital signal processing techniques are used in practice (without this being restrictive since an analog, or even passive, filtering could also be suitable). These techniques drive the coaxial multiway source and are for example implemented in a digital signal processor. They carry out for example an axial delay compensation, a cross-over filtration and an equalization enabling an apparent improvement of the field radiated by the coaxial multiway acoustic source. The crossover filters are also called bypass filters.
In order to achieve better control over certain residual defects (fluctuations in the radiation pattern or in the index of directivity at the overlap frequency bands) due to differences between the directivities proper to the different transducers, it has been proposed to optimize the cross-over filtering at the level of these overlap frequency bands. This optimization is based on a weighting of the frequency responses of the cross-over filters making it possible to modify the differences between the amplitudes and the phases of different channels of the source.
In order to resolve this problem of optimizing the cross-over filter, an overall approach is described in detail in the following documents:                the article (denoted [1]) “An optimized full-bandwidth 20 Hz-20 kHz digitally controlled co-axial source”, 5-8 Oct. 2006, San Francisco, Audio Engineering Society, Convention Paper (Shaiek, Debail, Kerneis, Boucher and Diquelou); and        thesis by M. Shaiek, defended on 2 Jul. 2007, “Optimisation des performances d′enceintes coaxiales large bande par traitement numérique du signal” (Optimizing wideband coaxial acoustic sets by digital signal processing”).        
In this overall approach, the filter synthesizing algorithm takes account, in a same cost function, of the different parameters to be optimized. This overall approach is based on an iterative digital search for complex weightings in using the gradient algorithm. The idea is to increase the cost so that in addition to the axial response and the radiation pattern, possible fluctuations of the directivity index of the source at the overlap frequency bands are minimized.
It may be recalled that the goal sought in implementing this optimizing of cross-over filtering according to the overall approach is to obtain a coaxial multiway acoustic source having an affine progression of its directivity index as a function of frequency, even at the overlap frequency bands.
This known technique (described in detail in the article and the dissertation mentioned here above) in no way seeks to obtain variable directivity. The target directivity therein is predetermined: it is fixed at the time of manufacture of the acoustic set and is therefore not adapted to the room or to the layout and to the user's preferences. This known technique does not provide for modifying the slope of the straight line representing the affine function of the directivity index nor does it vertically translate this straight line.
The overlap frequency bands between adjacent transducers (membranes) are discussed solely because the directivity index of the acoustic set undergoes undesired fluctuations (defects) in these overlap bands because of differences between the directivities proper to the different transducers.
Furthermore, these overlap frequency bands are of small widths compared with the frequency bands of the different transducers (membranes), and therefore the frequency band of operation of the acoustic set.
3.2.2 Other Cases of Acoustic Sources with Predefined and Non-Variable Directivity
The directivity of a speaker is created by interference between the signals generated at different points of the membrane of this speaker. To obtain a narrowing of the directivity pattern, the dimensions of the membrane need to be of the order of the wavelength of the signal sent (i.e. the sound signal generated by the membrane). This makes it necessary to have wide-diameter membranes at the bottom of the spectrum.
In stereo listening, the acoustic sets have a directivity pattern that is fixed at the time of their design. In a reverberating environment, one solution for diminishing the relative importance of the reverberation is to bring the acoustic sets closer to the listening position (in order to favor the direct wave/reflected wave ratio). But bringing the distance between the acoustic set and the listener into play amounts to setting constraints for the user. These are constraints which the user generally cannot apply except by modifying the furnishing and configuration of the listening room. In a reverberating environment, another solution to diminish the relative size of the reverberation consists in laying out absorbent elements in the room (in order to reduce the reverberation). However such a mode of processing the acoustics of the room is not simple to implement and is not necessarily possible. In a highly sound-dampened environment, the only parameter of adjustment that can be used to try and obtain a desired direct wave/reflected wave ratio remains the orientation of the axes of these acoustic sets; This can be done so as to favor reflections if any in order to bring early reflections towards the listener.
In multichannel listening, the rear acoustic sets can be dipolar. This technique consists in making two speakers work back to front, in phase opposition, so as to present a figure-of-eight type of directivity in order to create diffused sound without direct wave scattering. However, di-polar acoustic sets do not enable the direct wave/reflected wave ratio to be modulated.
In multichannel listening, another solution to obtain a desired pattern of directivity is to use an acoustic set comprising two identical speakers placed one behind the other and to create a phase shift between these two speakers (by applying a delay to one of the two speakers). This old principle, known as the “gradient speaker” principle is limited to the field of the low frequencies since, for higher frequencies, the directivity is further dictated by the phenomena of diffraction of the acoustic set and by the intrinsic directivity of the speakers. Furthermore, with this principle, the directional effect is obtained solely by an effect of cancellation of phase of one speaker relative to the other. This entails high penalties in terms of acoustic efficiency. This principle makes it possible to approach only first-order directivities. Now, it is sometimes desired to obtain far more complex directivities.
3.2 Source with Variable Directivity
To generate a pattern of variable directivity, there are known solutions based on networks of speakers, filtered for example by means of a DSP. In this way, an “acoustic antenna” is constituted and the amplitude and the phase of each of the speakers (the speakers are all identical) is used. The directivity of the acoustic antenna is generated by means of a signal processing by controlling the signals sent to each of the speakers. The directivity in no way results from the intrinsic directivity of the membranes of the network of speakers forming the antenna since all the speakers (and therefore all the membranes) are identical but rather from their spatial positions which are different. Besides, the directivity can become variable only if the dimensions of the acoustic antenna are of the order of magnitude of the length of the wave to be reproduced. Super-directive algorithms exist but they entail penalties in terms of sensitivity and robustness of performance (the intrinsic disparities of the speakers will lead to an accentuated disparity of performance of the acoustic antenna, especially in terms of pattern of directivity).
One drawback of these solutions is that they make it necessary to have rear distance so that the acoustic set can work in a field known as a far field. For example, a network of speakers with a width of 4 cm works in fat field conditions at 6300 Hz when the listener is at more than 3 m.
Another drawback of these approaches is that it offers only the possibility of configuring the acoustic antenna (i.e. synthesizing a given directivity) at a given point in time but is not designed for dynamic modification (no directivity control signal) by the user or automatically.
Another drawback of these approaches is that they offer control of directivity only on one plane (in general the horizontal plane). Matrices of speakers are needed to enable management of directivity on another plane (the vertical plane for example), but the other planes passing through the axis of the system are more or less well controlled and necessarily different.
Yet another drawback of these approaches is that the range of frequencies of operation is limited since all the speakers of the network are identical and therefore must cover a wide frequency band. This is difficult or even impossible to obtain. In practice, the reproduced frequency band is therefore limited.